Many thanks for such an instant response. I'll adopt these ideas and try to
eliminate those DOFs related to bubble support following the workflow in
the tutorial.
Regards,
Lixing
On Tuesday, January 5, 2021 at 1:21:56 PM UTC+8 Wolfgang Bangerth wrote:
>
> Lixing,
>
> > I am trying to implement a stabiliazed weak form (e.g.
> advection-diffusion)
> > where the stabilization tensor is computed element-wise through a
> standard
> > bubble: \Pi(1-x_i^2). It seems that FE_Q_Bubbles should provides all I
> need,
> > but here are two things I am not quite clear about,
> >
> > 1. Is the definition of bubble function in FE_Q_Bubbles in [0,1] span or
> > [-1,1] span?
>
> Everything in deal.II is on the reference cell define as [0,1]^d.
>
>
> > Does it has the standard bubble shape (1 at element center and
> > vanishes at the edges)? The (2x_j-1)^{p-1} part is a little bit
> confusing to
> > me. The bubble function corresponds to linear element in FE_Q_Bubbles is
> the
> > standard bubble that I desire, but I am really confusing at the shape of
> this
> > expression at higher orders.
>
> For higher orders, you end up with multiple bubble functions, one for each
> j=0..dim-1. This wasn't quite clear from the documentation, and I'll
> submit a
> patch later for that.
>
>
> > 2. I tried to utilize this class (i.e., FE_Q_Bubbles). One issue is that
> it
> > takes the virtual nodes of the bubble function into account of the total
> DOFs,
> > which is not the way we prefer in the stabilization method.
>
> But the behavior is correct -- the class describes a finite element
> *space*
> and that space contains the bubble function. I think that what you are
> trying
> to do is to do static elimination of that degree of freedom right away,
> and
> that can be implemented as well but is not the philosophical view we
> generally
> take in deal.II if you select such an element.
>
> The question is what you want the finite element space to be (i) locally,
> on
> every cell, and (ii) globally. There are a number of tutorials that
> discuss
> these sorts of questions. I would encourage you to read through step-61
> and
> step-51, for example.
>
> Best
> WB
>
> --
> ------------------------------------------------------------------------
> Wolfgang Bangerth email: [email protected]
> www: http://www.math.colostate.edu/~bangerth/
>
>
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